Properties

Label 1368.2.a.g
Level $1368$
Weight $2$
Character orbit 1368.a
Self dual yes
Analytic conductor $10.924$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1368.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.9235349965\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{7} - 2 q^{11} + q^{13} + 5 q^{17} + q^{19} + q^{23} - 5 q^{25} + 3 q^{29} + 4 q^{31} + 2 q^{37} + 8 q^{41} - 8 q^{43} + 8 q^{47} + 2 q^{49} - 9 q^{53} - q^{59} + 14 q^{61} + 13 q^{67} - 10 q^{71} + 9 q^{73} - 6 q^{77} - 10 q^{79} - 10 q^{83} + 12 q^{89} + 3 q^{91} + 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 0 0 0 0 3.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(19\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1368.2.a.g 1
3.b odd 2 1 152.2.a.b 1
4.b odd 2 1 2736.2.a.k 1
12.b even 2 1 304.2.a.b 1
15.d odd 2 1 3800.2.a.d 1
15.e even 4 2 3800.2.d.f 2
21.c even 2 1 7448.2.a.g 1
24.f even 2 1 1216.2.a.l 1
24.h odd 2 1 1216.2.a.f 1
57.d even 2 1 2888.2.a.b 1
60.h even 2 1 7600.2.a.o 1
228.b odd 2 1 5776.2.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.a.b 1 3.b odd 2 1
304.2.a.b 1 12.b even 2 1
1216.2.a.f 1 24.h odd 2 1
1216.2.a.l 1 24.f even 2 1
1368.2.a.g 1 1.a even 1 1 trivial
2736.2.a.k 1 4.b odd 2 1
2888.2.a.b 1 57.d even 2 1
3800.2.a.d 1 15.d odd 2 1
3800.2.d.f 2 15.e even 4 2
5776.2.a.l 1 228.b odd 2 1
7448.2.a.g 1 21.c even 2 1
7600.2.a.o 1 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1368))\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7} - 3 \) Copy content Toggle raw display
\( T_{11} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 3 \) Copy content Toggle raw display
$11$ \( T + 2 \) Copy content Toggle raw display
$13$ \( T - 1 \) Copy content Toggle raw display
$17$ \( T - 5 \) Copy content Toggle raw display
$19$ \( T - 1 \) Copy content Toggle raw display
$23$ \( T - 1 \) Copy content Toggle raw display
$29$ \( T - 3 \) Copy content Toggle raw display
$31$ \( T - 4 \) Copy content Toggle raw display
$37$ \( T - 2 \) Copy content Toggle raw display
$41$ \( T - 8 \) Copy content Toggle raw display
$43$ \( T + 8 \) Copy content Toggle raw display
$47$ \( T - 8 \) Copy content Toggle raw display
$53$ \( T + 9 \) Copy content Toggle raw display
$59$ \( T + 1 \) Copy content Toggle raw display
$61$ \( T - 14 \) Copy content Toggle raw display
$67$ \( T - 13 \) Copy content Toggle raw display
$71$ \( T + 10 \) Copy content Toggle raw display
$73$ \( T - 9 \) Copy content Toggle raw display
$79$ \( T + 10 \) Copy content Toggle raw display
$83$ \( T + 10 \) Copy content Toggle raw display
$89$ \( T - 12 \) Copy content Toggle raw display
$97$ \( T - 14 \) Copy content Toggle raw display
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